“Some of the things I said were inappropriate and for that I apologise unreservedly,” Barton, 34, tweeted.

Barton added he did not feel staying away from training was necessary.

In a statement on his Twitter account on Thursday evening, the former Newcastle and Manchester City midfielder said he respected manager Mark Warburton’s decision.

He claimed “there were only words involved in the disagreement” during discussions at training on Tuesday.

“Nevertheless, some of the words used did overstep the mark,” said Barton.

Both Barton and midfielder Halliday, 24, played in Rangers’ 5-1 Old Firm defeat, the club’s biggest loss to their local rivals since going down 6-2 in August 2000.

The result prompted discussions which Barton claimed “involved some sharp disagreement”.

He added: “I regret what happened and on Monday I will report for training and I will do what it takes to help the team draw a line under it so that we can get back to the task at hand.

“I cannot, however, apologise for caring deeply about winning and for wanting to perform better myself and for Rangers to do much better.”

Moments after his Twitter statement, Barton published a tweet he later deleted reading: “Apologising doesn’t always mean that you’re wrong and the other person is right. It means you value your relationship more than your ego.”

At his Thursday media conference, Warburton said the media reaction to his side’s loss at Celtic Park had been overly negative, but made no mention of Barton’s absence from training.

Rangers, who refused to comment on the situation on Thursday, host Ross County on Saturday.

Barton joined Rangers on a two-year deal in the summer when his one-year contract at Burnley expired.

“I’m coming up here to be the best player in the country,” he announced on the day he signed for Rangers, who returned to the Scottish Premiership after four years outside the top tier.